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The models are ordered along the horizontal axis in order of increasing performance based on the proportion of characteristic-sorted cross-sections matched ; characteristics are ordered along the vertical axis in order of increasing matching difficulty measured as the fraction of all three-factor models able to match the return cross-section generated by sorting stocks on a given characteristic. Both the performance measure, and the frequency with which three-factor models match each cross-section are listed in parentheses along each axis.

Each cell on the figure is shaded black if the -factor model is able to match the cross-section based on characteristic ; shaded gray if the -factor model is unable to match the cross-section based on characteristic , and shaded white if factor model includes a factor constructed using characteristic. A few patterns are apparent. A few characteristics generated particularly challenging cross-sections of test portfolios, matched only by the few highest-ranked models.

Several characteristics are virtually impossible to reconcile with empirical three-factor models constructed using our procedure. The other characteristics seem to be more or less difficult to span depending on the subsample. Such lack of stability is consistent with the spurious nature of performance of many of the randomly constructed -factor models. The low degree of correlation in relative model performance across the two sub-samples is partly due to the sampling errors, but it also suggests that performance of many models in our set may be spurious.

Another possibility for data-mining is associated with the choice of the empirical procedure for return factor construction. Thus far we have used a straightforward procedure for constructing return factors as long-short portfolios of the top and bottom deciles of stocks sorted on each characteristic. One popular alternative approach, following Fama and French , prescribes a two-dimensional sort: first on firm size and then on a characteristic in case of Fama, the characteristic is the book-to-market ratio.

We apply a conceptually similar approach in our setting. Specifically, for each characteristic, we first sort firms into big and small big firms are above the median in market capitalization, small firms are below , form long-short portfolios within each size class, and then average the returns on the two long-short portfolios to construct a return factor. In Table 9, we report cross-sectional correlations of performance between the empirical factor models formed using our univariate factor construction method and the corresponding models with factors formed via the double-sorting procedure.

In Tables 10 and 11 we report very different top-twenty and bottom-twenty factor model lists compared to Tables 5 and 6. As an example, the model using net stock issues NSI and liquidity LIQ is the top twenty performing factor models in our original full-sample analysis Table 5 , but it is one of the worst-performing models over the full sample under the double-sorting method Table We can also compare overall factor model performance using the original one-dimensional sort factor construction Figure 3 panel A and the double-sort factor construction Figure 4.

While we observed in Table 9 a low correlation in model performance across the two factor construction methods, the relative predictability of characteristics is very similar. Similarly, investment-to-capital IK also appears to be spanned only by the highest-ranked models.

Finally, we examine the improvement in model performance caused by moving from three to four factors in the pricing models. We repeat our analysis by considering the universe of 2, four-factor models, consisting of the market portfolio and three -factors based on our list of 27 firm characteristics. We present the results for four-factor models in Appendix B. The best-performing four-factor model in Table B. Many of the twenty best-performing four-factor models add factors constructed on momentum MOM , standardized unexpected earnings SUE , investment over assets IA , and asset growth AG to one of the top-performing three-factor models.

All of these additions are based on characteristics that present the most challenge to the three-factor -models, as we show in Figure 3. Adding such factors to the three-factor models produces a slight mechanical improvement in performance by excluding the corresponding cross-section from the set of test portfolios. Beyond that, the improvement is minimal: most challenging cross-sections have little correlation with each other or with other -factors, and therefore it is not possible to capture many additional cross-sections by introducing a fourth -factor.

The potential hazards of data-mining are well known. Our findings show just how difficult it is to judge the performance of empirically constructed factor pricing models when both the return factors and the target cross-sections of assets are chosen in a virtually unrestricted manner. While the impressive performance of some of the models we consider is spurious, some models must indeed capture economically meaningful sources of risk.

Distinguishing one set from the other purely based on empirical performance seems difficult - if the factors included in a theoretically grounded risk-factor model are some of the many possible -factors, such a model is likely to be defeated in a pure performance horse-race by the spuriously picked champions.

Eugene F. Fama - Google Scholar Citations

The winner in such a horse-race is not necessarily a superior risk model. For example, the momentum factor MOM appears in at least one of the three best-performing three-factor models for the full sample, and each of the half-samples. Yet, without a convincing attribution of the return spread on the momentum-sorted portfolios to a well-understood source of risk, it is difficult to interpret momentum as a primitive risk factor of first-order economic importance. Other situations may be more ambiguous, and one may be able to offer at least a tentative ex-post theoretical justification for the top-performing model.

Such theory-mining can add a patina of false legitimacy to the spurious pricing models, exacerbating the effects of data-mining. For example, the top-performing model based on the standardized-unexpected-earnings SUE and the cashflow-to-price CP factors suggests some tantalizing possibilities for straddling the behavioral and neoclassical asset pricing paradigms to "motivate" a hybrid pricing model with empirical performance that is literally second to none. Needless to say, a superficial theory adds no more value than a spurious empirical result. In summary, our analysis lends further support to the notion that to distinguish meaningful pricing models from the spurious ones, we should place less weight on the number of seemingly anomalous return cross-sections the models are able to match, and instead closely scrutinize the theoretical plausibility and empirical evidence in favor or against their main economic mechanisms.

Table 1a. Table 1c. Table 2 presents results from a principal component analysis on the 27 characteristic-based return factors. The table shows the proportion of cumulative variation that the first principal components can capture. Results are presented over the whole sample period and subsamples and Table 3 presents factor loadings for the first three principal components extracted from the set of 27 factor returns.

Loadings are shown for the whole sample period and subsamples and Details on characteristic definitions and construction is in Appendix A. Table 4 presents results from regressing the characteristic-based return factors on the benchmark three-factor model, consisting of the market portfolio and the first two principal component vectors of the return factors.

The alpha coefficient, t-statistic, and from the regression is shown in the table for the whole sample period and subsamples and Table 5 lists the characteristic-based factors that constitute the top twenty linear factor models, in terms of the proportion of remaining characteristics they can capture, via the equal-weighted method.

Top factor models are shown for the whole sample period and subsamples and The universe of factor models is all three-factor models consisting of the market portfolio and two characteristic return factors C1, C2 from our list of Table 6 lists the characteristic-based factors that constitute the bottom twenty linear factor models, in terms of the proportion of remaining characteristics they can capture, via the equal-weighted method. Bottom factor models are shown for the whole sample period and subsamples and Table 7. Table 7 shows the rank correlation and correlation of factor model performance for the first subsample period versus the second subsample period HML : H igh m inus l ow factor that accounts for the spread between value and growth companies based on their book-to-market ratio.

Value stocks are companies with high book to value. Construction: Double sorting. First divide stocks into two sets based on market capitalisation. Then split each set into three based on their book-to-market ratio. The SMB is the average return of the three small portfolios minus the average return of the three big portfolios, i.

The HML is the average return of two value portfolios minus the average return of the two growth portfolios, i. Step 1 Let n denote the number of assets.